Hyperelliptic Jacobians and Hasse–Witt Matrix

Recall that a hyperelliptic curve is a smooth projective curve of genus $g \geq 2$ that admits a degree 2 morphism to the projective line $\mathbb{P}^1$. In this post, we will discuss how to compute the Hasse–Witt matrix of a hyperelliptic curve defined over a finite field, which encodes a lot of information about the curve’s Jacobian. I arrived at this topic by working on a problem related to the $L$-polynomials of hyperelliptic curves over finite fields in high genus.

Arithmetic of Hyperelliptic Curves

Recall a hyperelliptic curve $C/\mathbb F_q$ is a smooth projective curve of genus $\ge 2$ with a degree $2$ map to $\mathbb P^1$. Assume the characteristic of $\mathbb F_q$ is odd, it has affine model $y^2=f(x)$, where $f(x)$ is square free.

The Jacobian $\mathrm{Jac}(C)$ of a smooth projective curve $C$ is a principally polarizaed abelian variety, defined as the identity component the Picard scheme $\mathrm{Pic}^0(C)$, where the Picard functor is the fppf sheafification of the functor $(\mathrm{Sch}^{\mathrm{op}})_{\mathrm{fppf}}\to \mathbf{Ab}$

$$T\mapsto \mathrm{Pic}(C_T)/\pi_*\mathrm{Pic}(T)$$

representable by an abelian variety. A polarization of an abelian variety $A$ is an isogeny $\lambda:A\rightarrow A^{\vee}$, where $A^\vee:=\mathrm{Jac}_{A/k}$ is the dual abelian variety, arising from an ample line bundle in this way: for a line bundle $L$, define

$$\lambda_L:A\to A^\vee\qquad x\mapsto t^*_xL\otimes L^{-1}$$

where $t_x:A\rightarrow A$ is the translation map by $x$. Moreover it is called principal if it is an isomorphism. In the case of a Jacobian $J=\mathrm{Jac}(C)$, there is a principal polarization induced by the ample divisor $\Theta$, defined as the image of $C^{(g-1)}:=C^{g-1}/\mathrm{Sym}_{g-1}\to J$ that sends $D\mapsto \mathcal O_C(D-(g-1)P_0)$, where $P_0$ is a rational point, and $D$ is the divisor representation of an element in $C^{(g-1)}$.

In other words, the Jacobian of $C$ is the moduli space of the degree $0$ line bundles on a $C$. The Neron-Severi group of $C$ is defined to be $\mathrm{Pic}(C)/\mathrm{Pic}^0(C)$. There is a canonical exact sequence.

$$0\to \mathrm{Pic}^0(C)\rightarrow \mathrm{Pic}(C)\rightarrow \mathrm{NS}(C)\rightarrow 0$$

where $\mathrm{NS}(C)=\mathbb Z$ if $k$ is algebraically closed.

For a smooth projective curve $C$ of genus $g$, its Jacobian $J=\mathrm{Jac}(C)$ has dimension $g$.

Proof.

It suffice to show the tangent space $T_0\mathrm{Pic}_{C}=H^1(C, \mathcal O_C)$, then by Serre duality

$$\dim J=\dim T_0 J=\dim T_0 \mathrm{Pic}_C=\dim H^1(C, \mathcal O_C)=\dim H^0(C,\omega_C):=g$$

Let $C_{\varepsilon}:=C\times_{k}k[\varepsilon]/\varepsilon^2$. There is an exact sequence of sheaves

$$0\rightarrow \mathcal O_C\xrightarrow{f\mapsto 1+\varepsilon f} \mathcal O_{C_{\varepsilon}}^\times\xrightarrow{a+\varepsilon b\mapsto a}\mathcal O_{C}^\times\rightarrow 0$$

This induces long exact sequence

$$H^0\left(C, \mathcal{O}_{C_{\varepsilon}}^{\times}\right) \rightarrow H^0\left(C, \mathcal{O}_C^{\times}\right) \rightarrow H^1\left(C, \mathcal{O}_C\right) \longrightarrow H^1\left(C, \mathcal{O}_{C_{\varepsilon}}^{\times}\right)=\mathrm{Pic}(C_{\varepsilon}) \longrightarrow H^1\left(C, \mathcal{O}_C^{\times}\right)=\mathrm{Pic}(C)$$

which gives $H^1(C,\mathcal O_C)=\mathrm{Ker}(\mathrm{Pic}(C_\varepsilon)\rightarrow \mathrm{Pic}(C))=T_0\mathrm{Pic}_C$

For smooth projective curve $X$ over $\overline{k}$ with $\ell$ an invertible prime, $H^1(X,\mu_{\ell^n})=\mathrm{Pic}(X)[\ell^n]$.

Proof.

Start with Kummer sequence

$$1 \rightarrow \mu_{\ell^n} \rightarrow \mathbf{G}_m \xrightarrow{\ell^n} \mathbf{G}_m \rightarrow 1$$

Deriving cohomology

$$H^0\left(X, \mathbf{G}_m\right) \xrightarrow{\ell^n} H^0\left(X, \mathbf{G}_m\right) \rightarrow H^1\left(X, \mu_{\ell^n}\right) \rightarrow H^1\left(X, \mathbf{G}_m\right) \xrightarrow{\ell^n} H^1\left(X, \mathbf{G}_m\right)$$

Thus $H^1\left(X, \mu_{\ell^n}\right) \cong H^1(X,\mathbf{G}_m)[\ell^n]\cong \operatorname{Pic}(X)\left[\ell^n\right]$.

For a smooth projective curve $C$ of genus $g$ and its Jacobian $J=\mathrm{Jac}(C)$, we have an isomorphism of $\mathrm{Gal}_k$-modules

$$\mathrm{AJ}^*:H^1(C_{\overline{k}},\mathbb Q_\ell)\cong H^1(J_{\overline{k}}, \mathbb Q_\ell)$$

induced by the Abel-Jacobi map $\mathrm{AJ}:C\to J$ defined as $P\mapsto [P-P_0]$ with $P_0$ a rational point.

Proof.

We have $H^1\left(C, \mu_{\ell^n}\right) \cong \mathrm{Pic}(C)\left[\ell^n\right] \cong\mathrm{Pic}^0(C)\left[\ell^n\right] \cong J\left[\ell^n\right]$. Similarly, $H^1(J,\mu_{\ell^n})=\mathrm{Pic}^0(J)[\ell^n]=J^\vee [\ell^n]$ via canonical principal polarization. By the functoriality of the Kummer sequence, the pullback by Abel-Jacobi map $H^1\left(J, \mu_{\ell^n}\right) \rightarrow H^1\left(C, \mu_{\ell^n}\right)$ and the induced map by Abel-Jacobi map on torsion Picard functors $\mathrm{Pic}(J)[\ell^n]\to \mathrm{Pic}(C)[\ell^n]$ agree, which also agrees with $\mathrm{Pic}^0(J)[\ell^n]\to \mathrm{Pic}^0(C)[\ell^n]$, since torsion points have degree $0$. This map is inverse to the canonical polarization $\lambda_{\Theta}$ since $\mathrm{Jac}(C)=\mathrm{Alb}(C)$ using the Albanese universal property, hence an isomorphism. Therefore we obtain the desired isomorphism by passing to the limit.

The $L$-polynomial of a smooth algebraic curve $C/\mathbb F_q$ is the polynomial

$$L_C(T)=\mathrm{det}(1-TF|\mathrm{H^1}(C_{\overline{\mathbb F_q}},\mathbb Q_\ell))=\mathrm{det}(1-TF|\mathrm{H^1}(J_{\overline{\mathbb F_q}},\mathbb Q_\ell))$$

where $F=\mathrm{Frob}_q^{-1}$ is the geometric Frobenius, $\ell$ is an invertible prime in $C$, and $J=\mathrm{Jac}(C)$.

This polynomial appears in the numerator of the Weil zeta function

$$Z(C,T)=\mathrm{exp}\left(\sum_{n}\frac{|C(\mathbb F_{q^n})|}{n}T^n\right)=\frac{L_C(T)}{(1-T)(1-qT)}$$

by Weil's conjecture. Additionally, the $L$-polynomial (Weil polynomial) of an abelian variety determines it up to $\mathbb F_q$-isogeny. This is a very deep result of Honda-Tate theory.

Same setting as above, then

$$L_C(T)\equiv \det (1-T\mathrm{Frob}_q|J[\ell])\pmod{\ell}$$

Proof.

Write $J[\ell]$ in terms of the Tate module $T_\ell J=\varprojlim J[\ell^n]$, as $J[\ell]\cong T_\ell J/\ell T_\ell J $, which is a $\mathrm{Gal}_k$-module isomorphism. Therefore

$$\det (1-T\mathrm{Frob}_q|J[\ell])\equiv \det (1-T\mathrm{Frob}_q|T_\ell J/\ell T_\ell J )\equiv \det (1-T\mathrm{Frob}_q|T_\ell J )\pmod{\ell}$$

We show that $T_\ell J$ is the dual representation to $H^1(C_{\overline{\mathbb F_q}}, \mathbb Z_\ell)$ by showing $T_\ell J\cong H^1(C_{\overline{\mathbb F_q}}, \mathbb Z_\ell(1))$ and using Poincare duality, where we recall $\mathbb Z_\ell(1)=\varprojlim \mu_{\ell^n}$ is the Tate twist. By ⟦ref:lem-tors⟧, we have $J[\ell^n]=\mathrm{Pic}(C)[\ell^n]=H^1(C,\mu_{\ell^n})=H^1(C,(\mathbb Z/\ell^n)(1))$, and passing to the limit we have $T_\ell J\cong H^1(C,\mathbb Z_\ell (1))\cong H^1(C,\mathbb Z_\ell)^\vee$. Therefore a matrix over $T_\ell J$ can be identified as its inverse transpose over $H^1(C,\mathbb Z_\ell)^\vee$. Transpose does not change characteristic polynomial, so $\det(1-TF|H^1(C,\mathbb Z_\ell))=\det(1-T\mathrm{Frob}_q|T_\ell J)$, done.

Specifically, it turns out that it is very fruitful to consider the $L$-polynomial mod $2$ (recall we are assuming odd characteristics), as demonstrated by a paper by Costa–Donepudi–Fernando–Karemaker–Springer–West ⟦cite:CDFKSW22⟧ . We present their main theorem.

Let $C/\mathbb F_q$ be a hyperelliptic curve with $q$ odd and genus $g$. Let the partition $(d_1,...,d_r)$ of $2g+2$ be the cycle type of the permutation of Frobenius on the $2g+2$ geometric branch points of $C\to \mathbb P^1$, then we have

$$L_C(T)\equiv \frac{\prod_{i=1}^r(1-T^{d_i})}{(1-T)^2}\pmod{2}$$

Proof. (Sketch)

We have the standard result that $J[2]$ is $2g$-dimensional over $\mathbb F_2$. Let $W$ be the $2g+2$ geometric branch points, then we have a vector space $\mathbb F_2^W=\mathbb F_2^{2g+2}$ with a Frobenius action. Straightforwardly, the Frobenius can be represented as a block diagonal matrix over $\mathbb F_2^W$ with blocks of sizes $d_1,...,d_r$, each one a cyclic permutation matrix. One easily finds the characteristic polynomial to be $\prod_{i=1}^r(1-T^{d_i})\in\mathbb F_2[T]$. A standard result of Mumford ⟦cite:Mum07⟧ says: if $e\in J[2]$, then as a divisor either $e=\sum_{P\in U}P-|U|(\infty)$ (if $\infty$ is ramified) or $e=\sum_{P\in U}P-\frac{|U|}{2}(\infty_1+\infty_2)$ (if $\infty$ splits in two), for some subset $U\subseteq W$ of even size, where two such representations agree iff $U_1=U_2$ or $U_1=W\setminus U_2$. Let $Z=\langle (1,\dots,1) \rangle$ and $Z^\perp$ be the orthogonal space then it is the subspace of vectors of an even number of nonzero entries. Frobenius acts stably and also invertibly on the quotient $\mathbb F_2^W/Z^{\perp}$, so one finds the characteristic polynomial of the Frobenius on $Z^{\perp}$ to be $(1-T)^{-1}\prod_{i=1}^r(1-T^{d_i})$ via the exact sequence. Similarly by considering $Z^{\perp}/Z$, one finds the characteristic polynomial $(1-T)^{-2}\prod_{i=1}^r(1-T^{d_i})$, and voila.

Notably, this explains a numerical phenomenon when I was producing large amount of data. Still assuming odd characteristics:

Let $C/\mathbb F_q$ be a hyperelliptic curve of genus $g>2$ such that $L_C(T)=1+a_gT^g+q^gT^{2g}$, then $a_g$ is even.

Proof.

Since $q^n$ is odd $L_C(T)\equiv 1+T^g+T^{2g}\pmod{2}$. By ⟦ref:thm-hyp⟧ we have in $\mathbb F_2[T]$

$$(1+T)^2(1+T^g+T^{2g})=\prod_{i=1}^r(1+T^{d_i})$$

swapping negative signs with positive ones. The LHS has exactly two linear factors, and the RHS $1+T^{d_i}$ contributes one linear factor if $d_i$ is odd and two if $d_i$ is even. Thus, the number of Frobenius cycles $r$ can only be $1$ or $2$. For $g>2$, the LHS would need to have $6$ terms, contradiction.

The above analysis helps us understand the $L$-polynomial mod $\ell$, where $\ell\ne p$. However it does not help us understand it mod $p$. The Hasse-Witt matrix helps us understand the $L$-polynomial mod $p$.

Semilinear Algebra and Hasse–Witt Matrix

Let $\varepsilon$ be an automorphism of the field $K$, and write $a^{\varepsilon}:=\varepsilon(a)$ for all $a \in K$. A map $T: V \to W$ between two $K$-vector spaces is called $\varepsilon$-semilinear if it satisfies the following properties for all $v, w \in V$ and $a \in K$: $T(v + w) = T(v) + T(w)$ and $T(av) = a^{\varepsilon} T(v)$. Let $E=(e_1, \ldots, e_n)$ and $F=(f_1,\dots,f_n)$ be an ordered basis for $V$, we represent a $\varepsilon$-semilinear map $T$ by a matrix $[T]_{E}^{F}$. Unlike in linear algebra, we have $[Tv]_{F} = [T]_{E}^{F} [v]_{E}^{\varepsilon}$, where $[v]_{E}^{\varepsilon}$ is obtained by applying $\varepsilon$ to each entry of $[v]_{E}$, and change of basis is: $[T]_{F}^{F} = [\mathrm{id}]_{E}^{F} [T]_E^E([\mathrm{id}]_{E}^{F})^{\varepsilon}$. Iteration is given by $[T^n]_{E}^{E} = [T]_E^E ([T]_E^E)^{\varepsilon} \cdots ([T]_E^E)^{\varepsilon^{n-1}}$.

Let $V^*$ be the dual of $V$ and $E^*=(e^1, \ldots, e^n)$ be the corresponding dual basis. If $T: V \to W$ is a $\varepsilon$-semilinear map, then its adjoint $T^*: W^* \to V^*$ is a $\varepsilon^{-1}$-semilinear map and is characterized by the property that $\langle Tv, w \rangle = \langle v, T^* w \rangle^{\varepsilon^{-1}}$ for all $v \in V$ and $w \in W^*$. In terms of matrices, we have $[T^*]_{F^*}^{E^*} = ([T]_E^F)^{t,\varepsilon^{-1}}$, where $t$ denotes the transpose.

Let $k$ be a perfect field of characteristic $p > 0$, and let $C$ be a smooth projective curve over $k$. Let $F: C \to C$ be the absolute Frobenius morphism, which raises functions to their $p$-th power. Let $\sigma:k\rightarrow k$ be the Frobenius automorphism of $k$ with inverse $\tau$. The absolute Frobenius $F:C \to C$ is defined to be the identity on the underlying topological space of $C$ and sends $f\mapsto f^p$ on sections. Recall that the coherent cohomology $H^i(C,\mathcal O_C)$ is a $k$-vector space via the following process: the structure map $C\rightarrow \mathrm{Spec}(k)$ induces $k=H^0(\mathrm{Spec}(k),\mathcal O_{\mathrm{Spec}(k)})\rightarrow H^0(C,\mathcal O_C)$, and for each $f\in H^0(C,\mathcal O_C)$, the multiplication map $m_f:\mathcal O_C \to \mathcal O_C$ by $f$ induces $m_f^i: H^i(C,\mathcal O_C) \to H^i(C,\mathcal O_C)$, and $f\cdot \alpha := m_f^i(\alpha)$ for $\alpha \in H^i(C,\mathcal O_C)$ makes $H^i(C,\mathcal O_C)$ a $H^0(C,\mathcal O_C)$-module, and hence by composition also a $k$-vector space. The absolute Frobenius $F$ induces a $\sigma$-semilinear map $F^*: H^1(C,\mathcal O_C) \to H^1(C,\mathcal O_C)$.

Let $B$ be a basis of $H^1(C,\mathcal O_C)$, the Hasse–Witt matrix of $C$ with respect to $B$ is defined to be the matrix representation of the $\sigma$-semilinear map $F^*$ with respect to $B$, i.e. $[F^*]_B^B$.

Let $C/\mathbb F_q$ be a smooth projective curve. If $H$ is the Hasse-Witt matrix of $C$, then

$$L_C(T)\equiv \det(1-TH)\pmod p.$$

In other words, the characteristic polynomial computes the $L$-polynomial mod $p$.

Proof.

Either one uses the coherent Lefschetz trace formula or comparison with de Rham and crystalline cohomology, both are difficult. See Manin's original congruence ⟦cite:Man61⟧ .

The Hasse–Witt matrix contains a lot more information than just the $L$-polynomial mod $p$. In particular, the $p$-rank $f$ of $J$ (defined as $\# J[p](\overline{k})=p^f$) can be computed by $f=\mathrm{rank}(HH^{(p)}\cdots H^{(p^{g-1})})$, which implies that $J$ is ordinary iff $H$ is invertible.

Let $\omega_C$ be the sheaf of regular Kähler differentials on $C$. To compute Hasse-Witt matrices, we introduce another matrix which is closely related to the Hasse-Witt matrix called the Cartier-Manin matrix, to which end we need to define the Cartier operator, which is the Serre dual to Hasse-Witt. Recall that Serre duality gives a perfect pairing $H^0(C,\omega_C) \times H^1(C,\mathcal O_C) \to k$ defined by $(\omega, \alpha) \mapsto \mathrm{Tr}(\alpha \smile \omega)$, where $\smile$ is the cup product and $\mathrm{Tr}: H^1(C,\omega_C) \to k$ is the trace map. However, in service of our purposes, we shall use here a residue version of Serre duality: $(\omega, \alpha)\mapsto \sum_{P\in |C|}\mathrm{Res}(\alpha_P \omega)$, where $\alpha_P$ is the local principal part of $\alpha$ at $P$.

Fix $C$ a smooth projective geometrically integral curve over a field $k$, with function field $K=k(C)$. For each closed point $P\in |C|$, let $K_P=\mathrm{Frac}(\widehat{\mathcal O}_{C,P})$ be the fractional field of the completion of the local ring of $C$ at $P$. Let $t_P$ be the uniformizer in $\mathcal O_{C,P}$, then non-canonically, we have $\widehat{\mathcal O}_{C,P}=k(P)[[t_P]]$ and $K_P=k(P)((t_P))$.

With the setting above, define the adele ring of $C$ as the topological ring

$$\mathbb A_C=\prod_{P\in |C|}(K_P,\widehat{\mathcal O}_{C,P})$$

In other words it is a collection of $(f_P)_{P\in |C|}$ where only finitely many are allowed to have poles. For a divisor $D=\sum_{P}n_P P$, define

$$\mathbb A_C(D)=\prod_{P\in |C|}\widehat{\mathcal O_C(D)_P}=\{(f_P)_{P\in |C|}:\forall P\in |C|,\mathrm{ord}_P(f_P)+n_P\ge 0\}\subseteq\mathbb A_C$$

When $D=0$, we have $\mathbb A_C(0)=\prod_{P\in |C|}\widehat{\mathcal O}_{C,P}$ the integral adeles. The rational functions $K\to \mathbb A_C$ embeds diagonally into the adeles.

For each divisor $D=\sum_{P}n_P P$, there is a isomorphism of $k$-vector spaces $H^1(C,\mathcal O_C(D))\cong \mathbb{A}_C/(\mathbb{A}_C(D)+K)$.

Proof.

See ⟦cite:Ser88⟧ .

(Serre Duality)

The Serre pairing below is perfect

$$\begin{aligned}H^1(C,\mathcal O_C(D))\times H^0(C,\omega_C(-D))&\rightarrow k\\ ((f_P)_P,\omega) &\mapsto\sum_{P\in |C|}\mathrm{Res}_{P}(f_P\omega)\end{aligned}$$

where for a local meromorphic differential $f_P\mathrm{d}t_P=(a_{-m}t_P^{-m}+a_{-m+1}t_P^{-m+1}+\cdots)\mathrm{d}t_P$, the residue is $\mathrm{Res}_P(f_P\mathrm{d}t_P):=\mathrm{Tr}_{k(P)/k}(a_{-1})$.

Proof.

See ⟦cite:Ser88⟧ .

The Cartier operator $M:H^0(C,\omega_C) \to H^0(C,\omega_C)$ is defined as the semilinear adjoint of the induced map by the absolute Frobenius morphism $F^*: H^1(C,\mathcal O_C) \to H^1(C,\mathcal O_C)$ with respect to the Serre duality pairing, i.e. $\langle F^* \alpha, \omega \rangle = \langle \alpha, M\omega \rangle^{\tau}$ for all $\alpha \in H^1(C,\mathcal O_C)$ and $\omega \in H^0(C,\omega_C)$. The Cartier–Manin matrix with respect to a basis $B$ of $H^0(C,\omega_C)$ is the matrix of the Cartier operator with respct to $B$.

Concretely, we can derive a formula for the Cartier operator as follows.

Let $C/\mathbb F_q$ be a hyperelliptic curve with $q$ odd and genus $g$. For each closed point $P\in |C|$ with local parameter $t_P$, and some local meromorphic differential $\omega=(a_{-m}t^m_P+a_{-m+1}t^{m+1}_P+\cdots)\mathrm dt_P\in \widehat{\omega}_{C,P}\otimes_{\widehat{O}_{C,P}}k(P)((t_P))$, we have

$$M\omega=\sum_{j}a^{1/p}_{pj-1}t^{j-1}_P\mathrm dt_P$$

Proof.

By adjointness on an adeles supported only at one point with $f_P=f$, we have

$$\mathrm{Res}_P(f^p\omega)=\left(\mathrm{Res}_P(fM\omega)\right)^p$$

Suppose $M\omega=\left(\sum_{m}b_mt^m_P\right)\mathrm dt_P$, and take $f=t^r$, then $\mathrm{Res}_P(\left(\sum_{n}a_nt^{n+pr}_P\right)\mathrm dt_P)=\left(\mathrm{Res}_P(\left(\sum_{m}b_mt^{m+r}_P\right)\mathrm dt_P)\right)^p$. Computing both sides, we have $a_{-pr-1}=b^p_{-r-1}$ so $b_{-r-1}=a_{-pr-1}^{1/p}$ by perfect ground field assumption. I’m ignoring the complications with trace, but this is a technicality one can solve using nondegeneracy of trace pairing. Renaming indices $j:=-r$, then done.

Same setting as above. Let $y^2=f(x)$ be an affine model of $C$. Suppose $f(x)^{\frac{p-1}{2}}=\sum_{m}c_m x^m$, then with respect to the standard basis $\omega_i=x^{i-1}\frac{\mathrm dx}{y}$ for $i=1,\dots,g$, the Cartier–Manin matrix is $(c_{pi-j}^{1/p})_{1\le i,j\le g}$.

Proof.

Using ⟦ref:thm-local⟧, since $y^{p-1}=f(x)^{\frac{p-1}{2}}$

$$M\omega_i=M\left(x^{i-1}\frac{\mathrm dx}{y}\right)=M\left(x^{i-1}f(x)^{\frac{p-1}{2}}\frac{\mathrm dx}{y^p}\right)=\frac{1}{y}M(x^{i-1}f(x)^{\frac{p-1}{2}}\mathrm dx)=\frac{1}{y}M\left(\sum_{m}c_mx^{m+i-1}\mathrm dx\right)=\frac{1}{y}\sum_{j=1}^g c_{pj-i}^{1/p}x^{j-1}\mathrm dx=\sum_{j=1}^g c_{pj-i}^{1/p}\omega_j$$

and we are done.

Same setting as above. The dual basis of $H^1(C,\mathcal O_C)$ to the standard basis $H^0(C,\omega_C)$ is given by $\eta_i=-\frac{y}{2x^i}$ for $i=1,\dots,g$, with respect to which the Hasse-Witt matrix of $C$ is $(c_{pj-i})_{1\le i,j\le g}$.

Proof.

Straightforward by the semilinar adjunction between Hasse–Witt and Cartier–Manin matrices.

References

[Man61]Yuri I. Manin. The Hasse-Witt matrix of an algebraic curve. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 25(1). pp. 153--172. 1961. English translation in American Mathematical Society Translations, Series 2, Volume 45, pages 245--264, 1965. https://www.mathnet.ru/eng/im3370.
[AH19]Jeffrey D. Achter and Everett W. Howe. Hasse-Witt and Cartier-Manin matrices: A warning and a request. In Arithmetic Geometry: Computation and Applications. pp. 1--18. American Mathematical Society. 2019. doi:10.1090/conm/722/14534. arXiv:1710.10726.
[Ser88]Jean-Pierre Serre. Algebraic Groups and Class Fields. Graduate Texts in Mathematics. Vol. 117. Springer-Verlag. New York. 1988.
[Mum07]David Mumford. Tata Lectures on Theta II: Jacobian Theta Functions and Differential Equations. Modern Birkhäuser Classics. Birkhäuser Boston. 2007.
[CDFKSW22]Edgar Costa, Ravi Donepudi, Ravi Fernando, Valentijn Karemaker, Caleb Springer, and Mckenzie West. Restrictions on Weil polynomials of Jacobians of hyperelliptic curves. In Arithmetic Geometry, Number Theory, and Computation. pp. 259--276. Springer. 2022. doi:10.1007/978-3-030-80914-0_7. arXiv:2002.02067.